Liouville-Riemann-Roch theorems on abelian coverings

TL;DR

This paper extends Liouville-Riemann-Roch theorems to co-compact abelian coverings, combining classical divisor theory with solutions allowing poles at infinity, revealing intricate interactions between finite divisors and points at infinity.

Contribution

It develops a new framework for Liouville-Riemann-Roch theorems on abelian coverings, integrating divisor and infinity conditions, and explores their complex interactions.

Findings

Exact dimensions of solution spaces can be computed for periodic elliptic operators.

Results reveal significant similarities with classical index theorems.

Interaction between finite divisors and infinity is non-trivial and intricate.

Abstract

The classical Riemann-Roch theorem has been extended by N. Nadirashvili and then M. Gromov and M. Shubin to computing indices of elliptic operators on compact (as well as non-compact) manifolds, when a divisor mandates a finite number of zeros and allows a finite number of poles of solutions. On the other hand, Liouville type theorems count the number of solutions that are allowed to have a "pole at infinity." Usually these theorems do not provide the exact dimensions of the spaces of such solutions (only finite-dimensionality, possibly with estimates or asymptotics of the dimension. An important case has been discovered by M. Avellaneda and F. H. Lin and advanced further by J. Moser and M. Struwe. It pertains periodic elliptic operators of divergent type, where, surprisingly, exact dimensions can be computed. This study has been extended by P. Li and Wang and brought to its natural limit for the case of periodic elliptic operators on co-compact abelian coverings by P. Kuchment and Pinchover. Comparison of the results and techniques of Nadirashvili and Gromov and Shubin with those of Kuchment and Pinchover shows significant similarities, as well as appearance of the same combinatorial expressions in the answers. Thus a natural idea was considered that possibly the results could be combined somehow in the case of co-compact abelian coverings, if the infinity is "added to the divisor." This work shows that such results indeed can be obtained, although they come out more intricate than a simple-minded expectation would suggest. Namely, the interaction between the finite divisor and the point at infinity is non-trivial.